When I "Leave Animus" I am able to go outside and explore the modern day "Villa Auditore" as Desmond. There was an early mission to do some stuff out there, but otherwise I don't see much point and apparently if I stay outside too long something bad happens, though I don't know what - I've never waited.

I doubt this is a spoiler, but just in case:

What I have noticed is that there is a big stream of footprints that I can see with eagle vision. They lead from the stairs near the fountain directly to the spot where we have the animus set up. I've checked them out about a half dozen times, but I can't find any significance. Do they become important later - or do they basically have no importance?

1 Answer
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Like you, I spent some time trying to figure out what to do in modern Monteriggioni.

You can 2 two things:

the mission you're talking about

recover 5 hidden artifacts (with one you get the achievement Dust to Dust)

Aaaaand nothing else.

About your hidden box:

In modern-day Monteriggioni, a strange red trail was visible when Eagle Vision was activated. The trail led from the back entrance of the Villa Auditore to the fountain bearing the Assassin insignia. After the completion of Sequence 6, the trail was no longer visible. Darby McDevitt and Falko Poiker (two devs, i think) explained in an interview that the trail was intended to help players find their way back to the Sanctuary, though its red color was an oversight.